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Are Black Holes Elementary Particles?

Y.K. HaTemple University

2008

75 years since Solvay 1933

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In physics, there are two theoretical lengths

• Classical size

• Classical radius of an object given by its classical theory

• Quantum size

• Compton wavelength of a particle given by quantum mechanics

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Electron

• Classical radius: • Quantum length:

r

e

m c

2

2

2 4 2 1 0 1 0. cm2 8 2 1 0 1 3. cm

mc

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General Criterion

• If the classical radius of an object is larger than its Compton wavelength, then a classical description is sufficient.

• If the Compton wavelength of an object is larger than its classical size, then a quantum description is necessary.

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Black Holes

• Schwarzschild radius:

• Proportional to mass

• Compton wavelength:

• Proportional to inverse mass

RG M

c

22 mc

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Planck Mass

• At the Planck mass, the Schwarzschild radius is equal to the Compton wavelength and the quantum black hole is formed.

Mc

Gx gmP l

2 2 1 0 5.

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Planck Length

• Quantum black holes are the smallest and heaviest conceivable elementary particles. They have a microscopic size but a macroscopic mass.

lG

cx cmP l

33 31 6 1 0.

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Dual Nature• Quantum black holes are at the

boundary between classical and quantum regions.

• They obey the macroscopic Laws of Thermodynamics and they decay into elementary particles.

• They can have a semi-classical description.

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Quantum Gravity?

• There is a total lack of evidence of any quantum nature of gravity, despite intensive efforts to develop a quantum theory of gravity.

• Is is possible that quantum gravity is not necessary?

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In General Relativity

• Spacetime is a macroscopic concept.

• Is Einstein’s equation similar in nature to Navier-Stokes equation in fluid mechanics as a macroscopic theory?

ds g x dx dx2 ( )

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Nuclear Force

• Energy levels are quantized in nuclei, but nuclear force is not a fundamental force.

• The fundamental theory is Quantum Chromodynamics of quarks and gluons.

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Graviton

• A hypothetical spin-2 massless particle.

• The existence of the graviton itself in nature remains to be seen.

• At best it propagates in an a priori background spacetime.

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Wave Equation

• The gravitational wave equation, from which the graviton idea is developed, is inherently a weak field approximation in general relativity.

102

2

22

c th

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Detectability

• It is physically impossible to detect a single graviton of energy .

• Detector size has to be less than the Schwarzschild radius of the detector.

RRS

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Classical Gravity

• We take the practical point of view that gravitation is entirely a classical theory, and that general relativity is valid down to the Planck scale.

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Spacetime

• This means that spacetime is continuous as long as we are above the Planck scale.

• At the Planck scale, quantum black holes will appear and they act as a natural cutoff to spacetime.

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What is an elementary particle?

An elementary particle is a logical

construction.

• Are black holes elementary particles?

• Are they fermions or bosons?

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Present Goal

• To construct various fundamental quantum black holes as elementary particles, using the results in general relativity.

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Black Hole Theorems:

1. Singularity Theorem 1965

2. Area Theorem 1972

3. Uniqueness Theorem 1975

4. Positive Energy Theorem 1983

5. Horizon Mass Theorem 2005

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Horizon Mass Theorem

For all black holes: neutral, charged or rotating, the horizon mass is always equal to twice the irreducible mass observed at infinity.

Y.K. Ha, Int. J. Mod. Phys. D14, 2219 (2005)

M r M irr( ) 2

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Black Hole Mass

• The mass of a black hole depends on where the observer is.

• The closer one gets to the black hole, the less gravitational energy one sees.

• As a result, the mass of a black hole increases as one gets near the horizon.

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Asymptotic Mass

• The asymptotic mass is the mass of a neutral, charged or rotating black hole including electrostatic and rotational energy.

• It is the mass observed at infinity.

M

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Horizon Mass

• The horizon mass is the mass which cannot escape from the horizon of a neutral, charged or rotating black hole.

• It is the mass observed at the horizon.

M r( )

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Irreducible Mass

• The irreducible mass is the final mass of a charged or rotating black hole when its charge or angular momentum is removed by adding external particles to the black hole.

• It is the mass observed at infinity.

M irr

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Surprising Consequence !

• The electrostatic and the rotational energy of a general black hole are all external quantities.

• They are absent inside the black hole.

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Charged Black Hole

• A charged black hole does not carry any electric charges inside.

• Like a conductor, the electric charges stay at the surface of the black hole.

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Rotating Black Hole

• A rotating black hole does not rotate.

• It is the external space which is undergoing rotating.

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Significance of Theorem

• The Horizon Mass Theorem is crucial for understanding Hawking radiation.

Tc

kG M

3

8

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Energy Condition

• Black hole radiation is only possible if the horizon mass is greater than the asymptotic mass since it takes an enormous energy for a particle released near the horizon to reach infinity.

M r M( )

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Photoelectric Effect

• The incident photon must have a greater energy than that of the ejected electron in order to overcome binding.

maxkEhf

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Hawking Radiation

• No black hole radiation is possible if the horizon mass is equal to the asymptotic mass.

• Without black hole radiation, the Second Law of Thermodynamics is lost.

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Quantum Black Holes

• Mass - Planck mass

• Radius - Planck length

• Lifetime - stable & unstable

• Spin - integer & half-integer

• Type - neutral & charged

• Other - Area & intrinsic entropy

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Black Hole Types

Spin-0

unstable

Spin-1/2

unstable

Spin-1

unstable

Planck-charge

stable

M r M( ) M r M( )

M r M( ) M r M( )

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Spin-0

• A Planck-size black hole created in ultra-high energy collisions or in the Big Bang.

• Disintegrates immediately after it is formed and become Hawking radiation.

• Observable signatures may be seen from its radiation.

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Planck-Charge

• A Planck-size black hole carrying maximum electric charge but no spin.

• It is absolutely stable and cannot emit any radiation.

Q G MP l P l

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Spin-1/2

• A Planck-size black hole carrying angular momentum and charge

and magnetic moment .

• It is unstable and it will decay into a burst of elementary particles.

/ 2

3 2Q P l /

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Spin-1

• A Planck-size rotating black hole with angular momentum but no charge.

• It will also decay into a burst of elementary particles

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Micro Black Holes

• Microscopic black holes with higher mass and larger size may be constructed from the fundamental types.

G M

cG

Q

c

J

M

2 2 2

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Black Hole Area

AG M

c

Q

G M

J c

G M

G Q

c

81 1

4

2 2

4

2

2

2 2

2 4

2

4

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Quantization

• Quantization of the area of black holes is a conjecture, not a proof.

• Unphysical spins (transcendental and imaginary numbers) not found in quantum mechanics would appear.

• Integer and half-integer spins do not result in quantization of area.

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Ultra-High Energy Cosmic Rays

Theoretical Upper Limit

• K. Greisen, End to the Cosmic Ray Spectrum, Phys. Rev. Lett 16 (1966) 748

• G.T. Zatsepin and V.A. Kuzmin, Upper Limit of the Spectrum of Cosmic Rays, JETP Lett. 4 (1966) 78.

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GZK Effect

• Interaction of protons with cosmic microwave background photons would result in significant energy loss.

• Energy spectrum would show flux suppression above eV.6 1 0 1 9

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Cosmic Ray Experiments

AGASA

• A dozen events above GZK limit possibly detected.

Hi-Res

• GZK effect observed.

• There is no correlation with nearby sources.

Pierre-Auger

• GZK effect observed.

• Correlation with AGN sources.

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GZK Paradox

• Why are some cosmic ray energies theoretically too high if there are no near-Earth sources?

• Quantum black holes in the neighborhood of the Galaxy could resolve the paradox posed by the GZK limit on the energy of cosmic rays from distant sources.

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Annihilation

• Quantum black holes carrying maximum charges are absolutely stable.

• They can annihilate with opposite ones to produce powerful bursts of elementary particles in all directions with very high energies.

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Dark Matter

• Planck-charge quantum black holes could act as dark matter in cosmology without having to resort to new interactions and exotic particles because they are non-interacting particles.

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Planck-Charge Black Holes

• Their electrostatic repulsion exactly cancels their gravitational attraction.

• There is no effective potential between them at any distance.

• The net energy outside the black hole is identically zero.

• They behave like a non-interacting gas.

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Conclusion

• Quantum black holes could have a real existence and play a significant role in cosmology.

• They would be indispensable to understanding the ultimate nature of spacetime and matter.

• Their discovery would be revolutionary

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Gerard `t Hooft, Of fabulous fame.

Ploughing the quantum field,He set it aflame.

When those gauge particles, Leaping from virtual to real.

Telling the Yang-Mills saga,It is a dream come true.